The contribution of the present work is to constructively prove that there is a non-stationary substitution process that identifies the location of the root that can be statistically consistently estimated from data. Indeed, the model is shown to be consistently estimable for two taxa. This is not possible for general non-stationary processes [1]}, [2]}, so we make the additional assumption that the process is strand-symmetric; that the process of evolution is identical on the sense and antisense strands of DNA. The conditions of the proof ensure that the process is non-stationary, and we show that a non-reversible, stationary model is not identifiable so cannot be consistently estimated. This observation sheds some light on the success of non-stationary processes and the failure of non-reversible, stationary processes at detecting the root in the literature.
